Abstract: The first half of the minicourse introduces the topological dynamics (expansivity, Shadowing Lemma, Anosov Closing Lemma, Markov approximations) and basic ergodic theory (multiple mixing) of hyperbolic sets, the technical centerpieces being, respectively, the Shadowing Theorem and the Hopf Argument. The second half presents new "Godbillon-Vey" invariants of maximal isotropic foliatons that produce astonishingly simple new proofs of classical rigidity results and constructions of contact Anosov flows that show, among other things, the subtlety of rigidity results of Benoist-Foulon-Labourie (for smooth invariant foliations) and Foulon (for entropy).

Ralf Spatzier

Title: Higher Rank and Rigidity of Group Actions

Abstract: This lecture series concerns "hyperbolic” actions of “higher rank” groups on compact manifolds and their rigidity properties. Examples of such actions come from commuting Anosov diffeomorphisms on tori, or the action of the diagonal subgroup of SL(n,R) on a compact quotient. We will discuss these and other examples. One core conjecture in this area, due Katok and Spatzier, asserts that all “irreducible” actions of this type are smoothly conjugate to actions of algebraic nature, as in the examples above. We will present some history, introduce tools developed and discuss recent progress by Rodriguez Hertz and Wang following work of Fisher, Kalinin and Spatzier. Time permitting, I will also touch on related developments, in particular on measure rigidity of homogeneous higher rank actions and smooth classification of actions of higher rank lattices.

I will also discuss related problems and applications of the core ideas, for example on factor maps of projective actions of higher rank lattices.

Yoshifumi Matsuda

Title: Bounded Euler number of actions of 2-orbifold groups on the circle

Abstract: Burger, Iozzi and Wienhard defined the bounded Euler number for a continuous action of the fundamental group of a connected oriented surface of finite type possibly with punctures on the circle. A Milnor-Wood type inequality involving the bounded Euler number holds and its maximality characterizes Fuchsian actions up to semiconjugacy. The definition of the bounded Euler number can be extended to actions of 2-orbifold groups by considering coverings. A Milnor-Wood type inequality and the characterization of Fuchsian actions also hold in this case. In this talk, we describe when lifts of Fuchsian actions of certain 2-orbifold groups, such as the modular group, are characterized by its bounded Euler number.

Katsutoshi Shinohara

Title: On the minimality of semigroup actions on the interval C^1-close to the identity

Abstract: We consider semigroup actions on the interval generated by two attracting generators.

A classical result of Duminy says that if the two generators are sufficiently C^2-close to the identity, then there is a restriction on the (forward) minimal set. Namely, it must be the whole interval.

In this talk, I discuss such a problem in the C^1-topology: The conclusion is that we can construct a counterexample to the corresponding problem in the C^1-topology. Starting from the backgrounds, I will elucidate the point where the importance of the difference of the regularity appears.